Basic Aspects of Ferroelectricity Induced by Noncollinear Alignment of Spins

Basic principles of ferroelectric activity induced by the noncollinear alignment of spins are reviewed. There is a fundamental reason why the inversion symmetry can be broken by certain magnetic order. This situation occurs when the magnetic order simultaneously involves ferromagnetic (F) and antiferromagnetic (A) counterparts, transforming under the spatial inversion I and time reversal T as IF=F and ITA=A, respectively. The incompatibility of these two conditions results in breaking the inversion symmetry, which manifests itself in the electric polarization P→. The noncollinear alignment of spins is one of examples of such coexistence of F and A. This coexistence principle imposes a constraint on possible dependencies of P→ on the directions of spins, which can include only “antisymmetric coupling” in the bond, P→ij·[ei×ej], and “single-ion anisotropy”, ei· Π→ ei. Microscopically, P→ij can be evaluated in the framework of superexchange theory. For the single Kramers doublet, this theory yields P→ij∼r→ij0, where r→ij0 is the spin-dependent part of the position operator induced by the relativistic spin-orbit coupling. r→ij0 remains invariant under spatial inversion, providing the microscopic reason why noncollinear alignment of spins can induce P→ even in centrosymmetric crystals. The symmetry properties of r→ij0 can be rationalized from the viewpoint of symmetry of Kramers states. Particularly, the commonly used Katsura–Nagaosa–Balatsky (KNB) rule P→∝ϵ→ji×[ei×ej] (ϵ→ji being the direction of the bond ij) can be justified only for relatively high symmetry of the bonds. The single-ion anisotropy vanishes for the spin 12 or if magnetic ions are located in inversion centers, thus severely restricting the applicability of this microscopic mechanism. The properties of multiferroic materials are reconsidered from the viewpoint of these principles. A particular attention is paid to complications caused by possible deviations from the KNB rule.